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### simple harmonic motion (SHM)

In physics, simple harmonic motion (SHM) is a periodic motion that is neither driven nor damped. An object in simple harmonic motion experiences a net force which obeys Hooke's law; that is, the force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction of the displacement.

A simple harmonic oscillator is a system which undergoes simple harmonic motion. The oscillator oscillates about an equilibrium position (or mean position) between two extreme positions of maximum displacement in a periodic manner. Mathematically, the motion of the oscillator can be described by means of a sinusoidal function such that the displacement x from the equilibrium position is given by:

$x(t) = A\cos\left(\omega t + \varphi\right),$

where A is the amplitude, ω is the angular frequency such that ω = 2πf where f is the frequency in units of hertz, and φ is the phase which is the elapsed fraction of wave cycle in radians.

The angular frequency of the motion is determined by the intrinsic properties of the system (often the mass of the object and the force constant), while the amplitude and phase are determined by the initial conditions (displacement and velocity) of the system. The kinetic and potential energies of the system are in turn determined by both intrinsic properties and initial conditions.

Simple harmonic motion. In this moving graph, the vertical axis represents the coordinate of the particle (x in the equation), and the horizontal axis represents time (t).

Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration.

Simple harmonic motion provides the basis of the characterization of more complicated motions through the techniques of Fourier analysis.

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